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This question was born out of a discussion Is the real number structure unique? on Math SE, but since it is more philosophical than mathematical I decided to ask here.

From Gödel completeness and incompleteness theorems, assuming the standard axioms of math being consistent, the set of natural numbers N could look very different. However, in some sense, there seems to be the one we prefer, namely, the one that consists of all numbers that we can get by repeatedly adding 1 together; any other numbers would be considered non-standard, so to speak.

However, in the absence of an external world that allows us to define rigorously the meaning of "repeatedly adding 1 together", is there in some sense one true set of natural numbers? What are some of the known arguments for and against there being only one true set of natural numbers?

Related question At what order of logic do we have a unique model of the natural numbers? addresses a similar issue, but note that this one use second-order arithmetic, which is dependent on some sort of meta-theory in the external world.

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    See Categoricity for an interesting discussion. – Mauro ALLEGRANZA Apr 23 '17 at 19:47
  • There is a sense in which you're assuming certain positions in the philosophy of mathematics prima facie when you ask this question. A platonist might argue that there is one true arithmetic because the natural numbers are actual, real, abstract objects and therefore statements about them are factually true. To a platonist, it doesn't matter if mathematical logic or some other foundational theory can't capture all of the truths. One argument for the view would be a platonist argument that "the natural numbers actually exist" Arguments against that position would be anti-platonist arguments. – Not_Here Apr 24 '17 at 18:53
  • Your inquiry purports a grammatical confusion given your use of "true" - nothing more than a pun whether you ask regarding "one true set" or "one true arithmetic". – Mr. Kennedy Apr 24 '17 at 23:12
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In short: The so-called definition of natural numbers as those that can be obtained from 0 by adding 1 repeatedly is circular, but there is no viable alternative, which already makes it impossible to uniquely pin down natural numbers mathematically. Worse still, there does not seem to be ontological reason for believing in the existence of a perfect physical representation of any collection that satisfies PA under a suitable interpretation.


Why the so-called definition is circular

It is circular because "repeatedly" cannot be defined without essentially knowing natural numbers. You cannot use the natural numbers to do counting because you have not defined them yet! You are stuck; you must already know what are natural numbers before you can talk about iteration. This is why in mathematical logic the meta-system must already have the collection of natural numbers to be able to define what it means for a formal system to be arithmetically sound (prove only arithmetical sentences that are true in the 'true natural numbers'). And for a slightly less brief account of various assumptions needed in the foundations of mathematics, see this post.

Why there is no viable alternative

The problem comes right at the beginning even before you can talk about arithmetic. Think about just pure propositional logic. What is a well-formed formula in propositional logic? To even define that, you necessarily must assume the existence of finite strings in the real world, otherwise you cannot even have a precise syntactic form for sentences, not to even say a full logic system. Furthermore, it is not enough to have a constructivistic view of finite strings in the sense that you can recognize them while not necessarily generating them. This is because propositional logic has deduction rules that permit deducing "A ⇒ A ∨ A" for any propositional formula "A", which clearly implies that one must be able to generate arbitrarily long strings. Effectively, to even describe the deductive system for propositional logic one already must accept the existence of the collection of finite strings.

One might naively think that perhaps we do not need formal systems at all. But the only way we can precisely and objectively describe something to another person is by a finite sequence of symbols in a common language. Pictures do not work because they are subject to interpretation unless they are in an agreed fixed format, in which case they could easily be encoded by symbol strings anyway. And if we want to have logical reasoning, the mere notion of mathematical proof involves finite sequences of symbols, hence by accepting any formal system whatsoever as being meaningful, we already must accept the basic properties of string manipulation, which amount to accepting TC (the theory of concatenation). But TC (despite having just the concatenation operation and no arithmetic operations) is essentially incomplete, so we cannot pin down even the finite strings!

So we do not even have hope of giving to anyone a description that uniquely identifies a collection of finite strings, which naturally precludes doing the same for natural numbers. This fact holds under very weak assumptions, such as those required to prove Godel's incompleteness theorems. If one rejects those... Well one reason to reject them is the following...

Why there is no apparent physical model of PA

As far as we know in modern physics, one cannot store finite strings in any physical medium with high fidelity beyond a certain length, for which I can safely give an upper bound of 2^10000 bits. This is not only because the observable universe is finite, but also because a physical storage device with extremely large capacity (on the order of the size of the observable universe) will degrade faster than you can use it.

So description aside, we do not have any reason to even believe that finite strings have actual physical representation in the real world. This problem cannot be escaped by using conceptual strings, such as iterations of some particular process, because we have no basis to assume the existence of a process that can be iterated indefinitely, pretty much due to the finiteness of the observable universe, again.

Therefore we are stuck with the physical inability to even generate all finite strings, or to generate all natural numbers in a physical representation, even if we define them using circular natural-language definitions!


Further facts

There are two curious facts related to this. Firstly, despite the fact that PA (Peano arithmetic) is based on the assumption of an infinite collection of natural numbers, which as explained above cannot have a perfect physical representation, PA still generates theorems that seem to be true at least at human scales. My favourite example is HTTPS, whose decryption process relies crucially on the correctness of Fermat's little theorem applied to natural numbers with length on the order of thousands of bits. So there is some truth in PA at human scales.

This may even suggest one way to escape the incompleteness theorems, because they only apply to deterministic formal systems that roughly speaking have certain unbounded closure properties (see this paper about self-verifying theories for sharp results regarding the incompleteness phenomenon). Perhaps the physical 'fuzziness' due to quantum mechanics or the spacetime limitations may permit the real world to be governed by some kind of system that does is syntactically complete, but anyway such systems will not have arithmetic in full as we know it!

Secondly, any meta-system MS that can reason about finite strings and sets of finite strings can prove the incompleteness theorems about itself, which immediately implies that if MS is consistent then MS' = MS + ¬ Con(MS) is consistent but Σ1-unsound (from the perspective of MS). But think about it: How do you know that MS is not already Σ1-unsound? The unsoundness could be buried deeper as well. MS'' = MS + ¬ ω-Con(MS) is ω-consistent but of course not arithmetically sound.

The thing is that we don't have any way to decide whether or not our preferred meta-system (whether ZFC or something nice and predicative like ACA) is in fact arithmetically sound, until we actually find a proof of say "⬜..⬜(0=1)" for some number (possibly zero) of "⬜"s. We cannot just say that if nobody has found such a proof then it is good enough evidence that they do not exist, since Godel's speed-up theorem and elegant examples by Harvey Friedman show that it is possible for the shortest such proof to be so long as to be impossible for humans to discover by trial and error.

  • I was waiting for someone to post a list of counterpoint, but at this point I guess I should just accept this. Very informative, thank you. – carrotomato May 1 '17 at 14:20
  • @carrotomato: You're welcome! Feel free to come to the logic chat-room for more discussion if you wish, especially if you're interested in the technical details. =) – user21820 May 4 '17 at 13:19
  • @carrotomato: I recently proved a constructive separation theorem about the arithmetical hierarchy, namely that we can write a computer program, which when given as input any proof verifier program of a formal system that is Σ[n]-sound will output a proof verifier program of a formal system that is Σ[n]-sound but Σ[n+1]-unsound. Thus we can never be sure that a system is arithmetically sound even if we know that it is sound at some finite level. I thought you might be interested. – user21820 Jan 27 '18 at 4:42
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If you are unwilling to accept on faith some sort of meta-theory (which has to be stronger than the Peano arithmetic itself) the answer is no, the "intended interpretation", a.k.a. the "standard interpretation", of arithmetic is ephemeral. But this should not be surprising, your own phrasing "namely, the one that consist of all numbers that we can get by repeatedly adding 1 together; any other numbers would be considered non-standard", i.e. only these and nothing else, can not be formalized in the first order logic. At a minimum you need the second-order arithmetic, and if you formalize such "intuitions" there (i.e. assume "full semantics"), then that's the "meta-theory" required to prove existence and uniqueness (up to isomorphism) of a well-founded model of Peano arithmetic, see discussion under Why do we know that Gödel sentences are true in the standard model of set theory, but do not know if the continuum hypothesis is? But even with all the meta-theory one wants, there is no one "standard model" of set theory (ZFC), so one should wonder how much stock to put into the alleged "intuitions" about arithmetic as well.

I do not see however, how external world is of much help in "rigorously defining" addition of units. The usual view of those who see mathematical objects as non-fictions (whether platonists or intuitionists) is that arithmetic, and mathematics in general, is eternal/a priori, and while it can be applied empirically it depends on no such application. The "meta-theory" in such views is seen as formalizing either platonist mindsight (Frege, Gödel) or constructive intuition of the subject (Poincare, Brouwer), not anything external. Even Hilbert, who was a fictionalist about most of mathematics, still admitted such intuitions about symbols in meta-theory, see Was there a Kantian influence on Hilbert's formalist programme? Indeed, the whole idea of "one true arithmetic", and set-theoretic models of formal theories in general, is platonist in origin. At the root it is based on the naive, but psychologically powerful, stereotype that nouns talked about in mathematics actually refer to something out there, the way desks and chairs do. Already intuitionists weaken this stereotype to a point where their meta-theory is insufficient to prove that the "standard interpretation", or indeed any complete interpretation, of arithmetic exists. By denying the law of excluded middle they explicitly assert the opposite.

  • what happens if we discard model-theoretic semantics? then there is no interpreration at all, let alone "standard interpretation". constructively, there is no "adding"; Z is Nat, and Sn is Nat whenever n is Nat. We do not need to give any additional "meaning" to this unless we have already committed to a particular semantic conception of meaning involving language and matalanguage. but i don't need an interpretation to know what Sn means. – user20153 Apr 23 '17 at 20:11
  • maybe i'm repeating your argument in different words? – user20153 Apr 23 '17 at 20:12
  • @mobileink Yes, I think this is the usual underpinning, although I can imagine salvaging some incomplete "standard interpretation" based on proof-theoretic semantics combined with some quasi-Kantian notion of intuition. I once even asked why model-theoretic semantics is so entrenched despite its philosophical flaws, see Why is Tarski's notion of logical validity preferred to deductive one? – Conifold Apr 23 '17 at 20:22
  • are you familiar with Etchemendy's "Logical Consequence"? Blew my head off. It's not undisputed but he makes a very strong argument that the model-theoretic notion of consequence is just as bad as the derivational one. but as you mentioned, MT consequence is so thoroughly entrenched. a young scholar would risk career by challenging it. – user20153 Apr 23 '17 at 20:28
  • the really interesting point is that the informal concept of logical consequence now is ananologous to the informal concept of effective procedure circa 1930. Turing solved the latter, but we're still waiting for somebody to do same with former. – user20153 Apr 23 '17 at 20:34
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There are an infinite number of topologies on the real line; so if you think that the line should be considered together with its topology then one already has an infinite number of possibilities.

This is a question that has been posed by Zeno, Aristotle and more lately by Weyl.

A different way of looking at this question is through Category Theory; for example Synthetic Differential Geometry takes as an axiom that there is a number d such that it's square vanishes; this axiomatises a notion in the traditional calculus of Newton (his fluxions) where one merely ignores these quantities in a limiting operation (a notion understood by Aristotle by the way); it turns that this model of the real line has no points in Set; and all the functions on it is smooth.

(Algebraically, they're also known as the system of dual numbers; and are actually used in physics as super-symmetry where they model bosons and fermions together, for example in the super-space formalism).

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